Evolution equation with fractional Schr\"odinger operators: monotonicity and exponential decay of solutions in Morrey spaces

We consider evolution equation with fractional Schr\"odinger operators in Morrey spaces. We prove order preserving properties of the associated semigroup in Morrey scale. We prove monotonicity of the semigroup with respect to Morrey's potentials and give some precise estimates of its exponential growth. We show that Arendt and Batty's type condition on the potential is necessary for exponential decay of Morrey's norms of the semigroup and find a large class of dissipative potentials for which it is also sufficient.